Connection with PDEs Diffusion approximation cours ARO07–MSSD #5 Random Models of Dynamical Systems Introduction to SDE’s Connection with partial differential equations Fran¸ cois Le Gland INRIA Rennes + IRMAR people.rennes.inria.fr/Francois.Le_Gland/insa-rennes/ 2 December 2020, via Zoom 1 / 39
Connection with PDEs Diffusion approximation Connection with PDEs Diffusion approximation 2 / 39
Connection with PDEs Diffusion approximation this connection between ‚ a second–order partial differential equation (PDE) ‚ and a stochastic differential equation (SDE) works both ways ‚ provides a probabilistic representation for the solution of a PDE, in terms of the solution of a SDE, and makes it possible to design numerical Monte Carlo approximation schemes relying of this probabilistic representation ‚ provides a PDE satisfied by statistics (probability distribution, probability of some event, moment, Laplace transform, etc.) of the solution of a SDE generalizes the method of characteristics, a connection between ‚ a first–order partial differential equation (PDE) ‚ and an ordinary differential equation (ODE) 2 / 39
Connection with PDEs Diffusion approximation Introduction: method of characteristics consider the ordinary differential equation 9 X p t q “ b p X p t qq with time–independent coefficient: ‚ a d –dimensional drift vector b p x q defined on R d it is assumed that the global Lipschitz and linear growth conditions hold associated with this ODE is the first–order partial differential operator d b i p¨q B ÿ M “ B x i i “ 1 such that M f p x q “ f 1 p x q b p x q 3 / 39
Connection with PDEs Diffusion approximation let u p t , x q be the unique (and ’regular enough’) solution to the PDE B u for any p t , x q in r 0 , T q ˆ R d B t p t , x q ` M u p t , x q “ 0 for any x in R d u p T , x q “ φ p x q Theorem 1 u p t , x q “ φ p X t , x p T qq where X t , x p s q denote the solution at time t ď s ď T of the ODE starting from x P R d at time t 4 / 39
Connection with PDEs Diffusion approximation Proof the chain rule yields dt u p s , X t , x p s qq “ B u d B t p s , X t , x p s qq ` u 1 p s , X t , x p s qq b p X t , x p s qq “ B u B t p s , X t , x p s qq ` M u p s , X t , x p s qq “ 0 since B u B t p s , y q ` M u p s , y q “ 0 for any y P R d , and the identity holds in particular for y “ X t , x p s q therefore, the mapping s ÞÑ u p s , X t , x p s qq is constant, and in particular its value for s “ t is the same as its value for s “ T , hence u p t , x q “ u p t , X t , x p t qq “ u p T , X t , x p T qq “ φ p X t , x p T qq l 5 / 39
Connection with PDEs Diffusion approximation consider the equation ż t ż t X p t q “ X p 0 q ` b p X p s qq ds ` σ p X p s qq dB p s q 0 0 with a m –dimensional Brownian motion B “ p B p t q , t ě 0 q , and time–independent coefficients: ‚ a d –dimensional drift vector b p x q defined on R d ‚ a d ˆ m diffusion matrix σ p x q defined on R d it is assumed that the global Lipschitz and linear growth conditions hold associated with this SDE is the second–order partial differential operator d d B 2 b i p¨q B ÿ ÿ ` 1 L “ a i , j p¨q 2 B x i B x i B x j i “ 1 i , j “ 1 it is also assumed that the d ˆ d symmetric matrix a p x q “ σ p x q σ ˚ p x q satisfies the uniform ellipticity condition: there exist a positive constant µ ą 0 such that, for any vector ξ P R d and any point x P R d it holds d a i , j p x q ξ i ξ j “ ξ ˚ a p x q ξ ě µ | ξ | 2 ÿ i , j “ 1 6 / 39
Connection with PDEs Diffusion approximation Cauchy initial–value problem let u p t , x q be the unique (and ’regular enough’) solution of the PDE B u for any p t , x q in r 0 , T q ˆ R d B t p t , x q ` L u p t , x q ´ c p t , x q u p t , x q “ f p t , x q for any x in R d u p T , x q “ φ p x q Theorem 2 ż T u p t , x q “ E t , x r φ p X p T qq exp t´ c p s , X p s qq ds us t ż T ż s ´ E t , x r f p s , X p s qq exp t´ c p r , X p r qq dr u ds s t t 7 / 39
Connection with PDEs Diffusion approximation Remark introducing the solution u p t , x q of the PDE B u for any p t , x q in r 0 , T q ˆ R d B t p t , x q ` L u p t , x q “ 0 for any x in R d u p T , x q “ φ p x q yields Markov semigroup u p t , x q “ E t , x r φ p X p T qqs “ p Q p T ´ t q φ qp x q Remark introducing the solution u p t , x q of the PDE B u for any p t , x q in r 0 , T q ˆ R d B t p t , x q ` L u p t , x q ´ λ c p t , x q u p t , x q “ 0 for any x in R d u p T , x q “ 1 yields Laplace transform ż T u p t , x q “ E t , x r exp t´ λ c p s , X p s qq ds us t 8 / 39
Connection with PDEs Diffusion approximation Remark [duality] recall that ż t x µ p t q , f y “ x µ p 0 q , f y ` x µ p s q , L f y ds 0 or in some sense B B t µ p t q “ µ p t q L the mapping ż t ÞÑ x µ p t q , u p t qy “ µ p t , dx q u p t , x q E is constant, i.e. does not depend on the time variable 0 ď t ď T 9 / 39
Connection with PDEs Diffusion approximation PDE interpretation dt x µ p t q , u p t qy “ x B B t µ p t q , u p t qy ` x µ p t q , B d B t u p t qy “ x µ p t q L , u p t qy ` x µ p t q , ´ L u p t qy “ 0 probabilistic interpretation E r φ p X p T qqs “ E r E r φ p X p T qq | F p t qs s “ E r E r φ p X p T qq | X p t qs s ż “ µ p t , dx q E r φ p X p T qq | X p t q “ x s E ż “ µ p t , dx q u p t , x q “ x µ p t q , u p t qy E 10 / 39
Connection with PDEs Diffusion approximation Proof (let 0 ď t ď T be fixed throughout the proof) introducing the process ż s V p s q “ exp t´ c p r , X p r qq dr u t the usual chain rule yields d dt V p s q “ ´ c p s , X p s qq V p s q or in integrated form ż s V p s q “ 1 ´ c p r , X p r qq V p r q dr t since the solution u is ’regular enough’, the Itˆ o formula yields ż s rB u u p s , X p s qq “ u p t , X p t qq ` B t p r , X p r qq ` L u p r , X p r qqs dr t ż s u 1 p r , X p r qq σ p X p r qq dB p r q ` t 11 / 39
Connection with PDEs Diffusion approximation the integration by parts formula for the process u p s , X p s qq V p s q yields ż s rB u u p s , X p s qq V p s q “ u p t , X p t qq ` B t p r , X p r qq ` L u p r , X p r qqs V p r q dr t ż s V p r q u 1 p r , X p r qq σ p X p r qq dB p r q ` t ż s ` u p r , X p r qq r´ c p r , X p r qq V p r qs dr t collecting all the ordinary integral terms reduces to ż s rB u B t p r , X p r qq ` L u p r , X p r qq ´ c p r , X p r qq u p r , X p r qqs V p r q dr t ż t “ f p r , X p r qq V p r q dr s since B u B t p r , y q ` L u p r , y q ´ c p r , y q u p r , y q “ f p r , y q for any y P R d , and the identity holds in particular for y “ X p r q 12 / 39
Connection with PDEs Diffusion approximation therefore ż s u p s , X p s qq V p s q “ u p t , X p t qq ` f p r , X p r qq V p r q dr t ż s V p r q u 1 p r , X p r qq σ p X p r qq dB p r q ` t taking s “ T and taking expectation (assuming the integrand belongs to M 2 pr 0 , T sq so that the stochastic integral has zero expectation) yields ż T u p t , x q ` E t , x r f p r , X p r qq V p r q dr s “ E t , x r u p T , X p T qq V p T qs t “ E t , x r φ p X p T qq V p T qs l 13 / 39
Connection with PDEs Diffusion approximation Dirichlet boundary–value / initial–boundary–value problems let D be an bounded open connected subset of R d , with smooth boundary B D #1 let u p x q be the unique (and ’regular enough’) solution of the PDE L u p x q ´ c p x q u p x q “ f p x q for any x in D u p x q “ φ p x q for any x on B D Theorem 3 introducing the stopping time τ “ inf t t ě 0 : X p t q R D u if such time exists, and τ “ 8 otherwise ż τ u p x q “ E 0 , x r φ p X p τ qq exp t´ c p X p s qq ds us 0 ż τ ż t ´ E 0 , x r f p X p t qq exp t´ c p X p s qq ds u dt s 0 0 14 / 39
Connection with PDEs Diffusion approximation Remark introducing the solution u p x q of the PDE L u p x q “ ´ 1 for any x in D u p x q “ 0 for any x on B D yields mean exit time u p x q “ E 0 , x r τ s Remark introducing the solution u p x q of the PDE L u p x q ´ λ u p x q “ 0 for any x in D u p x q “ 1 for any x on B D yields Laplace transform of exit time u p x q “ E 0 , x r exp t´ λ τ us 15 / 39
Connection with PDEs Diffusion approximation Remark introducing the solution u p x q of the PDE L u p x q “ 0 for any x in D u p x q “ φ p x q for any x on B D yields moment of exit position u p x q “ E 0 , x r φ p X p τ qqs 16 / 39
Connection with PDEs Diffusion approximation the challenge is to apply the Itˆ o formula to a function (the solution u of the PDE) that is defined only in D , and whose regularity at the boundary B D is not guaranteed introducing ‚ the open ε –interior subset D ε “ t x P D : d p x , B D q ą ε u of D ‚ a ’regular enough’ proxy function w ε defined on R d and which coincides on D ε with the solution u of the PDE ‚ the hitting time τ ε “ inf t t ě 0 : X p t q R D ε u if such time exists, and τ ε “ `8 otherwise it is then possible to apply the Itˆ o formula to the ’regular enough’ proxy function w ε , until the stopping time τ ε 17 / 39
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